Extension of $\Bbb Q_p$ is always abelian extension?

Question

Extension of $\Bbb Q_p$ is always abelian extension ?

I know the number of $p$-extension degree abelian extension of $\Bbb Q_p$ is $p+1$ from class field theory,but what about $p$-extension degree extension of $\Bbb Q_p$?

Answer 1

As explained in the comments, it might not always be abelian.

However, it will always be solvable, thanks to the theory of higher ramification groups (see e.g. Local Fields by Serre):

The Galois group $G$ of any (finite) Galois extension $L/K$ of local fields admits a filtration $G = G_{-1} \supseteq G_0 \supseteq G_1 \supseteq G_2 \supseteq \cdots$ of normal subgroups (in $G$), where:

• $G_0$ corresponds to the maximal unramified subextension $K^{nr}/K$ of $L/K$, so $G_{-1}/G_0$ is cyclic.
• $G_0 / G_1$ corresponds to the maximal tamely ramified subextension $K^t/K^{nr}$ of $L/K^{nr}$, so $G_0/G_1$ is cyclic with order coprime to $p$.
• $G_n / G_{n+1}$ is a product of cyclic groups of order $p$ for $n \ge 1$.

So in particular each subquotient $G_n / G_{n+1}$ is cyclic, which is the next best thing after $G$ being abelian.